Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
437 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #24988 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #13544 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #13499 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13472 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13313 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #13068 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13017 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12916 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #12826 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3407 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#472 | C22 | 1 | 2 | yes | — |
| 13 | #2948 | C22 | 1 | 2 | yes | — |
| 14 | #2161 | C22 | 1 | 1 | yes | characteristic |
| 15 | #2153 | C22 | 1 | 2 | yes | — |
| 16 | C2×#361 | C22 | 1 | 2 | yes | — |
| 17 | C2×#361 | C22 | 1 | 2 | yes | — |
| 18 | #2140 | C22 | 1 | 2 | yes | — |
| 19 | #2113 | C22 | 1 | 1 | yes | characteristic |
| 20 | #2081 | — | 2 | 2 | no | — |
| 21 | #2048 | — | 2 | 2 | no | — |
| 22 | #1978 | C22 | 1 | 2 | yes | — |
| 23 | #1953 | C22 | 1 | 2 | yes | — |
| 24 | #1904 | — | 2 | 2 | no | — |
| 25 | #1833 | C22 | 1 | 2 | yes | — |
| 26 | #1829 | — | 2 | 2 | no | — |
| 27 | #1814 | C22 | 1 | 2 | yes | — |
| 28 | #1757 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1389 | C22 | 1 | 2 | yes | — |
| 30 | #1385 | C22 | 1 | 2 | yes | — |
| 31 | #1384 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1382 | C22 | 1 | 2 | yes | — |
| 33 | #1381 | C22 | 1 | 1 | yes | characteristic |
| 34 | #1379 | C22 | 1 | 2 | yes | — |
| 35 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 36 | C22×D8 | C23 | 1 | 2 | yes | — |
| 37 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 38 | #518 | — | 2 | 2 | no | — |
| 39 | C2×D16 | — | 2 | 4 | no | — |
| 40 | C2×D16 | — | 4 | 1 | no | — |
| 41 | #504 | — | 4 | 2 | no | — |
| 42 | C2×M32 | — | 4 | 1 | no | — |
| 43 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×M32 | C23 | 1 | 2 | yes | — |
| 45 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 46 | #472 | — | 2 | 4 | no | — |
| 47 | #472 | — | 4 | 2 | no | — |
| 48 | #442 | — | 2 | 4 | no | — |
| 49 | C4×D8 | — | 4 | 2 | no | — |
| 50 | #430 | — | 2 | 2 | no | — |
| 51 | C2×#108 | C23 | 1 | 2 | yes | — |
| 52 | C2×#108 | C23 | 1 | 2 | yes | — |
| 53 | #427 | C23 | 1 | 2 | yes | — |
| 54 | #421 | — | 2 | 2 | no | — |
| 55 | #418 | — | 2 | 2 | no | — |
| 56 | #418 | — | 2 | 2 | no | — |
| 57 | #413 | D4 | 1 | 1 | yes | characteristic |
| 58 | C2×#100 | — | 2 | 2 | no | — |
| 59 | C2×#100 | — | 2 | 2 | no | — |
| 60 | #410 | D4 | 1 | 1 | yes | characteristic |
| 61 | #405 | C23 | 1 | 2 | yes | — |
| 62 | #368 | — | 4 | 1 | no | — |
| 63 | #364 | — | 4 | 2 | no | — |
| 64 | #361 | — | 4 | 2 | no | — |
| 65 | #361 | — | 4 | 2 | no | — |
| 66 | #357 | — | 4 | 2 | no | — |
| 67 | C22⋊C16 | — | 4 | 1 | no | — |
| 68 | #344 | — | 2 | 2 | no | — |
| 69 | #344 | — | 2 | 2 | no | — |
| 70 | #344 | — | 2 | 2 | no | — |
| 71 | #344 | D4 | 1 | 2 | yes | — |
| 72 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 73 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C22×D4 | — | 2 | 2 | no | — |
| 75 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 76 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 77 | C2×SD16 | — | 4 | 2 | no | — |
| 78 | C2×D8 | — | 2 | 2 | no | — |
| 79 | C2×D8 | — | 2 | 2 | no | — |
| 80 | C2×D8 | — | 2 | 2 | no | — |
| 81 | C2×D8 | — | 2 | 2 | no | — |
| 82 | C2×D8 | — | 2 | 2 | no | — |
| 83 | C2×D8 | — | 4 | 2 | no | — |
| 84 | C2×D8 | #41 | 1 | 4 | yes | — |
| 85 | #131 | — | 2 | 2 | no | — |
| 86 | #131 | — | 2 | 2 | no | — |
| 87 | C2×M16 | — | 2 | 1 | no | — |
| 88 | C2×M16 | — | 2 | 2 | no | — |
| 89 | C2×M16 | — | 2 | 2 | no | — |
| 90 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 92 | C22×C8 | — | 2 | 1 | no | — |
| 93 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 94 | C4×D4 | — | 4 | 4 | no | — |
| 95 | C4×D4 | — | 8 | 2 | no | — |
| 96 | #117 | — | 2 | 2 | no | — |
| 97 | #117 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 99 | D16 | — | 4 | 4 | no | — |
| 100 | M32 | — | 2 | 2 | no | — |
| 101 | M32 | — | 2 | 2 | no | — |
| 102 | M32 | — | 2 | 2 | no | — |
| 103 | M32 | — | 4 | 2 | no | — |
| 104 | C2×C16 | — | 2 | 2 | no | — |
| 105 | C2×C16 | — | 2 | 1 | no | — |
| 106 | C2×C16 | — | 2 | 1 | no | — |
| 107 | C2×C16 | — | 2 | 2 | no | — |
| 108 | C2×C16 | — | 4 | 2 | no | — |
| 109 | #108 | — | 2 | 2 | no | — |
| 110 | #108 | — | 2 | 2 | no | — |
| 111 | #108 | — | 2 | 2 | no | — |
| 112 | #108 | — | 2 | 2 | no | — |
| 113 | #108 | — | 8 | 2 | no | — |
| 114 | #107 | — | 2 | 2 | no | — |
| 115 | #106 | — | 2 | 2 | no | — |
| 116 | #104 | — | 8 | 2 | no | — |
| 117 | #102 | — | 4 | 4 | no | — |
| 118 | #102 | — | 8 | 2 | no | — |
| 119 | #101 | — | 2 | 1 | no | — |
| 120 | #100 | — | 2 | 1 | no | — |
| 121 | #100 | — | 4 | 4 | no | — |
| 122 | #100 | — | 8 | 2 | no | — |
| 123 | #99 | — | 4 | 1 | no | — |
| 124 | #97 | — | 2 | 2 | no | — |
| 125 | C4×C8 | — | 2 | 2 | no | — |
| 126 | C24 | — | 4 | 2 | no | — |
| 127 | #41 | — | 2 | 2 | no | — |
| 128 | #41 | — | 2 | 2 | no | — |
| 129 | C2×Q8 | #135 | 1 | 1 | yes | characteristic |
| 130 | C2×D4 | — | 2 | 1 | no | — |
| 131 | C2×D4 | — | 2 | 4 | no | — |
| 132 | C2×D4 | — | 4 | 2 | no | — |
| 133 | C2×D4 | — | 4 | 2 | no | — |
| 134 | C2×D4 | — | 4 | 2 | no | — |
| 135 | C2×D4 | — | 4 | 2 | no | — |
| 136 | C2×D4 | — | 4 | 2 | no | — |
| 137 | C2×D4 | #135 | 1 | 1 | yes | characteristic |
| 138 | C22×C4 | — | 2 | 1 | no | — |
| 139 | C22×C4 | — | 8 | 4 | no | — |
| 140 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 141 | SD16 | — | 8 | 4 | no | — |
| 142 | D8 | — | 2 | 4 | no | — |
| 143 | D8 | — | 2 | 4 | no | — |
| 144 | D8 | — | 4 | 2 | no | — |
| 145 | D8 | — | 4 | 2 | no | — |
| 146 | D8 | — | 8 | 4 | no | — |
| 147 | M16 | — | 2 | 2 | no | — |
| 148 | M16 | — | 2 | 2 | no | — |
| 149 | M16 | — | 2 | 2 | no | — |
| 150 | M16 | — | 4 | 2 | no | — |
| 151 | M16 | — | 4 | 2 | no | — |
| 152 | M16 | — | 4 | 2 | no | — |
| 153 | M16 | — | 4 | 2 | no | — |
| 154 | M16 | — | 4 | 2 | no | — |
| 155 | C2×C8 | — | 2 | 1 | no | — |
| 156 | C2×C8 | — | 2 | 2 | no | — |
| 157 | C2×C8 | — | 4 | 1 | no | — |
| 158 | C2×C8 | — | 4 | 1 | no | — |
| 159 | C2×C8 | — | 4 | 2 | no | — |
| 160 | C2×C8 | — | 4 | 2 | no | — |
| 161 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 162 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 163 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 164 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 165 | C4⋊C4 | — | 2 | 2 | no | — |
| 166 | C4⋊C4 | — | 4 | 2 | no | — |
| 167 | C22⋊C4 | — | 2 | 2 | no | — |
| 168 | C22⋊C4 | — | 4 | 2 | no | — |
| 169 | C22⋊C4 | — | 8 | 4 | no | — |
| 170 | C42 | — | 2 | 2 | no | — |
| 171 | C42 | — | 4 | 2 | no | — |
| 172 | C16 | — | 4 | 1 | no | — |
| 173 | C16 | — | 4 | 2 | no | — |
| 174 | C16 | — | 4 | 1 | no | — |
| 175 | C23 | — | 2 | 1 | no | — |
| 176 | C23 | — | 4 | 4 | no | — |
| 177 | C23 | — | 8 | 2 | no | — |
| 178 | C23 | — | 8 | 2 | no | — |
| 179 | C23 | — | 32 | 2 | no | — |
| 180 | C23 | #575 | 1 | 1 | yes | characteristic |
| 181 | Q8 | — | 2 | 2 | no | — |
| 182 | D4 | — | 2 | 2 | no | — |
| 183 | D4 | — | 4 | 2 | no | — |
| 184 | D4 | — | 4 | 4 | no | — |
| 185 | D4 | — | 4 | 4 | no | — |
| 186 | D4 | — | 8 | 2 | no | — |
| 187 | D4 | — | 8 | 2 | no | — |
| 188 | C2×C4 | — | 2 | 1 | no | — |
| 189 | C2×C4 | — | 4 | 1 | no | — |
| 190 | C2×C4 | — | 4 | 1 | no | — |
| 191 | C2×C4 | — | 4 | 2 | no | — |
| 192 | C2×C4 | — | 8 | 2 | no | — |
| 193 | C2×C4 | — | 32 | 4 | no | — |
| 194 | C2×C4 | #537 | 1 | 1 | yes | characteristic |
| 195 | C2×C4 | #546 | 1 | 2 | yes | — |
| 196 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 197 | C2×C4 | #576 | 1 | 2 | yes | — |
| 198 | C8 | — | 4 | 1 | no | — |
| 199 | C8 | — | 4 | 2 | no | — |
| 200 | C8 | — | 8 | 2 | no | — |
| 201 | C8 | — | 8 | 2 | no | — |
| 202 | C8 | #539 | 1 | 2 | yes | — |
| 203 | C8 | #551 | 1 | 1 | yes | characteristic |
| 204 | C8 | #560 | 1 | 1 | yes | characteristic |
| 205 | C22 | — | 4 | 1 | no | — |
| 206 | C22 | — | 4 | 1 | no | — |
| 207 | C22 | — | 8 | 1 | no | — |
| 208 | C22 | — | 8 | 4 | no | — |
| 209 | C22 | — | 16 | 4 | no | — |
| 210 | C22 | — | 32 | 2 | no | — |
| 211 | C22 | — | 32 | 2 | no | — |
| 212 | C22 | #3250 | 1 | 1 | yes | characteristic |
| 213 | C22 | #3319 | 1 | 2 | yes | — |
| 214 | C4 | — | 4 | 1 | no | — |
| 215 | C4 | — | 8 | 2 | no | — |
| 216 | C4 | — | 16 | 2 | no | — |
| 217 | C4 | #2693 | 1 | 1 | yes | characteristic |
| 218 | C4 | #3259 | 1 | 1 | yes | characteristic |
| 219 | C4 | #3297 | 1 | 2 | yes | — |
| 220 | C2 | — | 2 | 2 | no | — |
| 221 | C2 | — | 2 | 1 | no | — |
| 222 | C2 | — | 8 | 1 | no | — |
| 223 | C2 | — | 16 | 4 | no | — |
| 224 | C2 | #31555 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31555 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13313 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #24988 | C2 | semidirect |
| 2. #13544 | C2 | semidirect |
| 3. #13499 | C2 | semidirect |
| 4. #13472 | C2 | semidirect |
| 6. #13068 | C2 | semidirect |
| 7. #13017 | C2 | semidirect |
| 8. #12916 | C2 | semidirect |
| 9. #12826 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #24988 | 2 |
| #13544 | 2 |
| #13499 | 2 |
| #13472 | 2 |
| #13313 | 2 |
| #13068 | 2 |
| #13017 | 2 |
| #12916 | 2 |
| #12826 | 2 |